Eigenvalues of zero-divisor graphs of finite commutative rings

Abstract

We investigate eigenvalues of the zero-divisor graph Gamma (R) of finite commutative rings R and study the interplay between these eigenvalues, the ring-theoretic properties of R and the graph-theoretic properties of Gamma (R). The graph Gamma (R) is defined as the graph with vertex set consisting of all nonzero zero-divisors of R and adjacent vertices x, y whenever xy = 0. We provide formulas for the nullity of Gamma (R), i.e., the multiplicity of the eigenvalue 0 of Gamma (R). Moreover, we precisely determine the spectra of Gamma ({mathbb {Z}}_p times {mathbb {Z}}_p times {mathbb {Z}}_p) and Gamma ({mathbb {Z}}_p times {mathbb {Z}}_p times {mathbb {Z}}_p times {mathbb {Z}}_p) for a prime number p. We introduce a graph product times _{Gamma } with the property that Gamma (R) cong Gamma (R_1) times _{Gamma } cdots times _{Gamma } Gamma (R_r) whenever R cong R_1 times cdots times R_r. With this product, we find relations between the number of vertices of the zero-divisor graph Gamma (R), the compressed zero-divisor graph, the structure of the ring R and the eigenvalues of Gamma (R).